January  2015, 14(1): 185-199. doi: 10.3934/cpaa.2015.14.185

Mean value properties and unique continuation

1. 

Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain

Received  January 2014 Revised  March 2014 Published  September 2014

In the first part of the paper we review some mean value properties and their connections to the Laplacian and other significant nonlinear operators like the $p$-Laplacian and the infinity-Laplacian. The second part is devoted to the unique continuation property, including a brief description of the methods, some of the main problems in the area and connections to the so called infinity mean value property.
Citation: José G. Llorente. Mean value properties and unique continuation. Communications on Pure & Applied Analysis, 2015, 14 (1) : 185-199. doi: 10.3934/cpaa.2015.14.185
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show all references

References:
[1]

Ark. Mat., 6 (1967), 551-561.  Google Scholar

[2]

Ark. Math., 7 (1968), 395-425.  Google Scholar

[3]

Manuscripta Mathematica, 47 (1984), 133-151. doi: 10.1007/BF01174590.  Google Scholar

[4]

in Proc. Japan -United States Sem., Tokyo (1977), 1-6.  Google Scholar

[5]

Springer-Verlag, 1991. doi: 10.1007/b97238.  Google Scholar

[6]

Bull. of the American Mathematical Society (New series), 41 (2004), 439-505. doi: 10.1090/S0273-0979-04-01035-3.  Google Scholar

[7]

Ber. Ver. Sächs. Akad. Wiss. Leipzig, 68 (1916), 3-7. Google Scholar

[8]

in Some topics in nonlinear PDEs (Turin, 1989). Rend. Sem. Mat. Univ. Politec. Torino 1989, Special Issue, 1568, 1991.  Google Scholar

[9]

Ark. for Mat., 26B (1939), 1-9. Google Scholar

[10]

in Calculus of variations and nonlinear partial differential equations, Lecture Notes in Mathematics, 1927 (2008), 75-122. doi: 10.1007/978-3-540-75914-0_3.  Google Scholar

[11]

Interscience Publishers, 1962.  Google Scholar

[12]

Calc. Var. Partial Differential Equations, 13 (2001), 123-139.  Google Scholar

[13]

IEEE Trans. Image Processsing, 7 (1998), 376-386. doi: 10.1109/83.661188.  Google Scholar

[14]

Wadsworth Mathematics Series, 1984.  Google Scholar

[15]

(1840), Werke, 5, Band, Göttingen, 1877. Google Scholar

[16]

Indiana Univ. Math. J., 35 (1986), 245-268. doi: 10.1512/iumj.1986.35.35015.  Google Scholar

[17]

Expo. Math., 30 (2012), 154-167. doi: 10.1016/j.exmath.2012.01.006.  Google Scholar

[18]

Nonlinear Analysis, 101 (2014), 89-97. doi: 10.1016/j.na.2014.01.020.  Google Scholar

[19]

J. London Math. Soc., 29 (1954), 491-497.  Google Scholar

[20]

Acta Math., 171 (1993), 139-163. doi: 10.1007/BF02392531.  Google Scholar

[21]

J. London Math. Soc., 50 (1994), 349-360. doi: 10.1016/j.exmath.2008.04.001.  Google Scholar

[22]

Arch. Rational Mech. Anal., 123 (1993), 51-74. doi: 10.1007/BF00386368.  Google Scholar

[23]

Ann. of Math., 12 (1985), 463-494. doi: 10.2307/1971205.  Google Scholar

[24]

SIAM J. Math. Anal., 33 (2001), 699-717. doi: 10.1137/S0036141000372179.  Google Scholar

[25]

Trans. Amer. Math. Soc., 36 (1934), 227-242. doi: 10.2307/1989835.  Google Scholar

[26]

Journal des Mathéatiques Pures et Apliquées, 97 (2012) , 173-188. doi: 10.1016/j.matpur.2011.07.001.  Google Scholar

[27]

in Proceedings of the International Congress of Mathematics, (Berkeley 1986). Vol. 1,2. Amer. Math. Soc. (1987), 948-960.  Google Scholar

[28]

in Harmonic Analysis and Partial differential equations (El Escorial 1987). Lecture Notes in Math., 1384 (1989), 69-90. doi: 10.1007/BFb0086794.  Google Scholar

[29]

Sitzungsber. Berlin. Math. Gessellschaft, 5 (1906), 39-42. Google Scholar

[30]

Hath. Math. Monographs, 1968.  Google Scholar

[31]

Comm. on Pure and Appl. Math., 43 (1990), 127-136. doi: 10.1002/cpa.3160430105.  Google Scholar

[32]

Report, University of Jyväkylä Department of Mathematics and Statistics, 102 (2006).  Google Scholar

[33]

Nonlinear Differential Equations and Applications, 14 (2007), 29-55. doi: 10.1007/s00030-006-4030-z.  Google Scholar

[34]

Ann. Acad. Scient. Fennicae, 39 (2014), 473-483. doi: 10.5186/aasfm.2014.3914.  Google Scholar

[35]

Siam J. Math. Anal., 29 (1998), 279-292. doi: 10.1137/S0036141095294067.  Google Scholar

[36]

Differential Integral Equations, 3-4 (2014), 201-216.  Google Scholar

[37]

Proc. Amer. Math. Soc., 103 (1988), 473-479. doi: 10.2307/2047164.  Google Scholar

[38]

Proc. Amer. Math. Soc., 138 (2010), 881-889. doi: 10.1090/S0002-9939-09-10183-1.  Google Scholar

[39]

Ann. Sc. Norm. Super. Pisa Cl. Sc., 11 (2013), 215-241.  Google Scholar

[40]

in Classical and modern potential theory and applications. NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 430, Kluwer (1994), 359-398.  Google Scholar

[41]

Rec. Math. Moscou (Mat. Sbornik), 32 (1925), 464-471. Google Scholar

[42]

Duke Math. J., 145 (2008), 91-120. doi: 10.1215/00127094-2008-048.  Google Scholar

[43]

Journal American Math. Soc., 22 (2009), 167-210. doi: 10.1090/S0894-0347-08-00606-1.  Google Scholar

[44]

Marcel Dekker, 1994.  Google Scholar

[45]

Rend. Acadd. d. Lincei Roma, 18 (1909), 263-266. Google Scholar

[46]

Electronic J. of Differential Equations, 122 (2006), 1-4.  Google Scholar

[47]

Rend. Circ. Mat. Palermo, 19 (1905), 140-150. Google Scholar

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